Given X_1,X_2,…,X_n are i.i.d with f(x)=1/π 1/(1+x^2). Find the minimal sufficient statistic for 1/(π^n(∏^n_(i=1)1+(x_i−θ)^2). If we are given the entire sample set, we could easily obtain the order statistic of that sample, so how could T(X)=(x_1,…,x_n) is wrong?

duandaTed05

duandaTed05

Answered question

2022-10-13

Given X 1 , X 2 , , X n are i.i.d with f ( x ) = 1 π 1 1 + x 2 . Find the minimal sufficient statistic for 1 π n ( i = 1 n 1 + ( x i θ ) 2 ) .
If we are given the entire sample set, we could easily obtain the order statistic of that sample, so how could T ( X ) = ( x 1 , , x n ) is wrong?

Answer & Explanation

ohhappyday890b

ohhappyday890b

Beginner2022-10-14Added 12 answers

T ( X ) = X is a sufficient statistics but NOT minimal.
T ( X ) = X contains more information than T ( X ) = ( x ( 1 ) , x ( 2 ) , , x ( n ) ).
T ( X ) = X contains RANK while the order statistic does not. The order statistic only contains the values.
For example X = ( 1 , 2 , 3 , 4 , 5 ) and X = ( 1 , 3 , 2 , 4 , 5 ) both has the same order statistic T ( X ) = ( 1 , 2 , 3 , 4 , 5 ) ..

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